If one works from “patterns” alone, this is an easy slip to make, as 2 is the only exception, the only even prime. Students sometimes believe that all prime numbers are odd. This is a much more useful result than having every number be expressible as a product of primes in an infinite number of ways, so we define prime in such a way that it excludes 1. Excluding it leaves only these cases: 3 × 2 × 2 2 × 3 × 2 2 × 2 × 3 In fact, if we call 1 a prime, then there are infinitely many ways to write any number as a product of primes. Well, if we include 1, there are infinitely many ways to write 12 as a product of primes. Using 4, 6, and 12 clearly violates the restriction to be “using only prime numbers.” But what about these? 3 × 2 × 2 2 × 3 × 2 1 × 2 × 3 × 2 2 × 2 × 3 × 1 × 1 × 1 × 1 To understand why it is useful to exclude 1, consider the question “How many different ways can 12 be written as a product using only prime numbers?” Here are several ways to write 12 as a product but they don’t restrict themselves to prime numbers. Even the informal idea rules it out: it cannot be built by multiplying other (whole) numbers.īut why rule it out?! Students sometimes argue that 1 “behaves” like all the other primes: it cannot be “broken apart.” And part of the informal notion of prime - we cannot compose 1 except by using it, so it must be a building block - seems to make it prime. It says “two distinct whole-number factors” and the only way to write 1 as a product of whole numbers is 1 × 1, in which the factors are the same as each other, that is, not distinct. A formal definitionĪ prime number is a positive integer that has exactly two distinct whole number factors (or divisors), namely 1 and the number itself. To capture the idea that “7 is not divisible by 2,” we must make it clear that we are restricting the numbers to include only the counting numbers: 1, 2, 3…. The number 7 can be composed as the product of other numbers: for example, it is 2 × 3. That captures the idea well, but is not a good enough definition, because it has too many loopholes. Informally, primes are numbers that can’t be made by multiplying other numbers. Numbers like this are called prime numbers. To “build” 7, we must use 7! So we’re not really composing it from smaller building blocks we need it to start with.
For example, he only way to build 7 by multiplying and by using only counting numbers is 7 × 1. Some numbers can’t be built from smaller pieces this way. Numbers like 10 and 36 and 49 that can be composed as products of smaller counting numbers are called composite numbers. Prime and composite numbers: We can build 36 from 9 and 4 by multiplying or we can build it from 6 and 6 or from 18 and 2 or even by multiplying 2 × 2 × 3 × 3. But only some counting numbers can be composed by multiplying two or more smaller counting numbers. Building numbers from smaller building blocks: Any counting number, other than 1, can be built by adding two or more smaller counting numbers.
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